Ok, I think I got it.
My AI is in a foriegn language so the description may not be exact.
1) draw as many of the rounded rectangles as you like, group them together into one layer, chose the layer, all squares will be highlighted as one object
2) go to Object - Envelope Deformation - Make with warp or
with the object highligted press alt+shift+ctrl+w
3) in the little popup, first go and click the Preview box under OK and Cancel so you can see what's going on when changing the settings
4) in the first rolldown choose Blowup (circle, third from bottom)
5) I only changed the first percentage bar to 26%
Click OK and you should now have the same squares but the outer ones will be bulging out into a round shape. This is also called a "fisheye" effect, i.e. in photography.
6) goto Effect - 3D - rotate
7) again choose preview so you can see what's going on "live"
8) my settings are 64°, 40°, -19°
9) change perspective to about 114°
10) click OK!
Hope this helps